Answer
$\frac{\sqrt{2+\sqrt{2}}}{2}$
Work Step by Step
Use the half-angle formula, $\sin\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{2}}$. Note that $67.5^\circ$ is in Quadrant I, where sine is positive, so we take the positive square root.
$\sin 67.5^\circ$
$=\sin \frac{135^\circ}{2}$
$=\sqrt{\frac{1-\cos 135^\circ}{2}}$
$=\sqrt{\frac{1-\left(-\frac{\sqrt{2}}{2}\right)}{2}}$
$=\sqrt{\frac{\left(1+\frac{\sqrt{2}}{2}\right)*2}{2*2}}$
$=\sqrt{\frac{2+\sqrt{2}}{4}}$
$=\frac{\sqrt{2+\sqrt{2}}}{\sqrt{4}}$
$=\frac{\sqrt{2+\sqrt{2}}}{2}$
